The main goal of this work is to present the proof-theoretical and model-theoretical approaches to probabilistic logics which allow reasoning about independence and probabilistic support. We extend the existing formalisms [14] to obtain several variants of probabilistic logics by adding the operators for independence and confirmation to the syntax. We axiomatize these logics, provide corresponding semantics, prove that the axiomatizations are sound and strongly complete, and discuss decidability issues.

We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.